Former Research Group: Tropical geometry
The former Research Group "Tropical Geometry" was lead by Hannah Markwig.
In tropical geometry, algebraic varieties are degenerated to piece-wise linear objects called tropical varieties. Although information gets lost in the degeneration process, many properties remarkably continue to hold on the tropical side. Because of their piece-wise linearity tropical varieties can be studied using combinatorial or linear algebra methods. Thus tropical geometry is a new and promising tool to prove new algebraic geometry theorems. (For example, Welschinger invariants (that can be viewed as real analogues of Gromov-Witten invariants) can be computed tropically. There is no way known to compute them without tropical geometry.) Also combinatorics and discrete mathematics can benefit from this new connection to algebraic geometry. In addition, tropical geometry has many other applications, e.g. in biomathematics and computer science.
This picture shows two tropical cubics in the plane.
One possibility to define the degeneration process that yields a tropical variety from an algebraic variety is to pick a certain field over which we do algebraic geometry, namely the field of Puiseux series. A Puiseux series is a formal power series with complex coefficients and rational exponents that share a common denominator. There is a non-archimedean valuation map which sends a Puiseux series to the smallest exponent (and 0 to infinity). We denote the coordinate-wise valuation map (with reversed sign) by Val and define the tropicalization of an algebraic variety over the Puiseux series to be the closure its image under Val. The first surprise of tropical geometry is that this image is a piece-wise linear combinatorial object. The second surprise is that many properties of the algebraic variety can be read off from this combinatorial object, even though the map Val loses much information.
Besides using tropical geometry as an application for algebraic geometry or other fields, our aim is to study tropical geometry for itself to get a better understanding of this new and exciting field.
This picture shows a tropical plane in space.
Here are some recent succesful applications of tropical geometry in various fields of mathematics: